Itô’s formula

0.1 Case of single space dimension

Let $X_t$ be an Itô process satisfying the stochastic differential equation

$$d{X}_t={\mu }_{t}dt+{\sigma }_{t}d{W}_t,$$

with ${\mu }_t$ and ${\sigma }_t$ being adapted processes, adapted to the same filtration as the Brownian motion $W_t$. Let $f$ be a function with continuous partial derivatives $\frac{\partial f}{\partial t}$, $\frac{\partial f}{\partial x}$ and $\frac{{\partial }^{2}f}{\partial x^2}$.

Then $Y_t=f(X_t)$ is also an Itô process, and its stochastic differential equation is

$d{Y}_t$ $={\displaystyle \frac{\partial f}{\partial t}}dt+{\displaystyle \frac{\partial f}{\partial x}}d{X}_t+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^{2}f}{\partial x^2}}(d{X}_t)(d{X}_t)$

$=\left({\displaystyle \frac{\partial f}{\partial t}}+{\displaystyle \frac{\partial f}{\partial x}}{\mu }_t+{\displaystyle \frac{1}{2}}{\sigma }_t^2\right)dt+{\displaystyle \frac{\partial f}{\partial x}}{\sigma }_{t}d{W}_t,$

where all partial derivatives are to be taken at $(t,X_t)$.

0.2 Case of multiple space dimensions

There is also an analogue for multiple space dimensions.

Let $X_t$ be a ${ℝ}^n$-valued Itô process satisfying the stochastic differential equation

$$d{X}_t={\mu }_{t}dt+{\sigma }_{t}d{W}_t,$$

with ${\mu }_t$ and ${\sigma }_t$ being adapted processes, adapted to the same filtration as the $m$-dimensional Brownian motion $W_t$. ${\mu }_t$ is ${ℝ}^n$-valued and ${\sigma }_t$ is $L({ℝ}^m,{ℝ}^n)$-valued.

Let $f:{ℝ}^n\times ℝ\to ℝ$ be a function with continuous partial derivatives.

Then $Y_t=f(X_t)$ is also an Itô process, and its stochastic differential equation is

$d{Y}_t$ $={\displaystyle \frac{\partial f}{\partial t}}dt+(\mathrm{D}f)d{X}_t+\frac{1}{2}d{X}_t^{*}({\mathrm{D}}^{2}f)d{X}_t$

$={\displaystyle \frac{\partial f}{\partial t}}dt+(\mathrm{D}f){\mu }_{t}dt+(\mathrm{D}f){\sigma }_{t}d{W}_t+\frac{1}{2}d{W}_t^{*}{\sigma }_t^{*}({\mathrm{D}}^{2}f){\sigma }_{t}d{W}_t$

$={\displaystyle \frac{\partial f}{\partial t}}dt+(\mathrm{D}f){\mu }_{t}dt+(\mathrm{D}f){\sigma }_{t}d{W}_t+\frac{1}{2}\mathrm{tr}\left({\sigma }_t^{*}({\mathrm{D}}^{2}f){\sigma }_t\right)dt$

$=\left({\displaystyle \frac{\partial f}{\partial t}}+(\mathrm{D}f){\mu }_t+\frac{1}{2}\mathrm{tr}\left(({\sigma }_t{\sigma }_t^{*})({\mathrm{D}}^{2}f)\right)\right)dt+(\mathrm{D}f){\sigma }_{t}d{W}_t,$

where

• $\mathrm{tr}$ is the trace operation; ${}^{\ast }$ is the transpose

• $\mathrm{D}f$ is the derivative with respect to the space variables; its value is a linear transformation from $L({ℝ}^n,ℝ)$

• ${\mathrm{D}}^{2}f$ is the second derivative with respect to space variables; represented as the Hessian matrix

• the third line follows because $d{W}_t^{i}d{W}_t^j={\delta }_{ij}dt$.

The quadratic form $\mathrm{tr}({\sigma }_t{\sigma }_t^{*}{\mathrm{D}}^{2}f)dt$ represents the quadratic variation of the process. When ${\sigma }_t$ is the identity transformation, this reduces to the Laplacian of $f$.

Itô’s formula in multiple dimensions can also be written with the standard vector calculus operators. It is in the similar notation typically used for the related parabolic partial differential equation describing an Itô diffusion:

$$d{Y}_t=\left(\frac{\partial f}{\partial t}+{\mu }_t\cdot \nabla f+\frac{1}{2}\left(\nabla \cdot ({\sigma }_t{\sigma }_t^{*})\nabla \right)f\right)dt+({\sigma }_{t}d{W}_t)\cdot \nabla f.$$

Title	Itô’s formula
Canonical name	ItosFormula
Date of creation	2013-03-22 17:16:14
Last modified on	2013-03-22 17:16:14
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	13
Author	stevecheng (10074)
Entry type	Axiom
Classification	msc 60H10
Classification	msc 60H05
Synonym	Itô’s formula
Synonym	Itô’s chain rule
Synonym	Ito’s formula
Synonym	Ito’s lemma
Synonym	Ito’s chain rule
Related topic	ItosLemma2
Related topic	GeneralizedItoFormula